Lecture 62 of 76Week 9
Examples of measures constructed using RRT
Prof. Indrava Roy · IMSc
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- Measure Theory, IMSc (at NPTEL)
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All the lectures
- Finite Sets and Cardinality
- Infinite Sets and the Banach-Tarski Paradox - Part 1
- Infinite Sets and the Banach-Tarski Paradox - Part 2
- Elementary Sets and Elementary measure - Part 1
- Elementary Sets and Elementary measure - Part 2
- Properties of elementary measure - Part 1
- Properties of elementary measure - Part 2
- Uniqueness of elementary measure and Jordan measurability - Part 1
- Uniqueness of elementary measure and Jordan measurability - Part 2
- Characterization of Jordan measurable sets and basic properties of Jordan measure - Part 1
- Characterization of Jordan measurable sets and basic properties of Jordan measure - Part 2
- Examples of Jordan measurable sets- I
- Examples of Jordan measurable sets- II - Part 1
- Examples of Jordan measurable sets- II - Part 2
- Jordan measure under Linear transformations - Part 1
- Jordan measure under Linear transformations - Part 2
- Connecting the Jordan measure with the Riemann integral - Part 1
- Connecting the Jordan measure with the Riemann integral - Part 2
- Outer measure - Motivation and Axioms of outer measure
- Comparing Inner Jordan measure, Lebesgue outer measure and Jordan Outer measure
- Finite additivity of outer measure on Separated sets, Outer regularity - Part 1
- Finite additivity of outer measure on Separated sets, Outer regularity - Part 2
- Lebesgue measurable class of sets and their Properties - Part 1
- Lebesgue measurable class of sets and their Properties - Part 2
- Equivalent criteria for lebesgue measurability of a subset - Part 1
- Equivalent criteria for lebesgue measurability of a subset - Part 2
- The measure axioms and the Borel-Cantelli Lemma
- Properties of the Lebesgue measure: Inner regularity,Upward and Downwar Monotone convergence theorem, and Dominated convergence theorem for sets - Part 1
- Properties of the Lebesgue measure: Inner regularity,Upward and Downwar Monotone convergence theorem, and Dominated convergence theorem for sets - Part 2
- Lebesgue measurability under Linear transformation, Construction of Vitali Set -Part 1
- Lebesgue measurability under Linear transformation, Construction of Vitali Set - Part 2
- Abstract measure spaces: Boolean and Sigma-algebras
- Abstract measure and Caratheodory Measurability - Part 1
- Abstract measure and Caratheodory Measurability - Part 2
- Abstrsct measure and Hahn-Kolmogorov Extension
- Lebesgue measurable class vs Caratheodory extension of usual outer measure on R^d
- Examples of Measures defined on R^d via Hahn Kolmogorov extension - Part 1
- Examples of Measures defined on R^d via Hahn Kolmogorov extension - Part 2
- Measurable functions: definition and basic properties - Part 1
- Measurable functions: definition and basic properties - Part 2
- Egorov's theorem: abstract version
- Lebesgue integral of unsigned simple measurable functions: definition and properties
- Lebesgue integral of unsigned measurable functions: motivation, definition and basic properties
- Fundamental convergence theorems in Lebesgue integration: Monotone convergence theorem, Tonelli's theorem and Fatou's lemma
- Lebesgue integral for complex and real measurable functions: the space of L^1 functions
- Basic properties of L^1-functions and Lebesgue's Dominated convergence theorem
- L^1 functions on R^d: Egorov's theorem revisited (Littlewood's third principle)
- L^1 functions on R^d: Statement of Lusin's theorem (Littlewood's second principle), Density of simple functions, step functions, and continuous compactly supported functions in L^1
- L^1 functions on R^d: Proof of Lusin's theorem, space of L^1 functions as a metric space
- L^1 functions on R^d: the Riesz-Fischer theorem
- Various modes of convergence of measurable functions
- Easy implications from one mode of convergence to another
- Implication map for modes of convergence with various examples
- Uniqueness of limits across various modes of convergence
- Some criteria for reverse implications for modes of convergence
- Riesz Representation theorem- Motivation
- Basics on Locally compact Hausdorff spaces
- Borel and Radon measures on LCH spaces
- Properties of Radon measures and Lusin's theorem on LCH spaces
- Riesz Representation theorem - Complete statement and proof - Part 1
- Riesz Representation theorem - Complete statement and proof - Part 2
- Examples of measures constructed using RRT
- Theorems of Tonelli and Fubini- interchanging the order of integration for repeated integrals: motivation and discussion of product measure spaces
- Product measures
- Tonelli's theorem for sets - Part 1
- Tonelli's theorem for sets - Part 2
- Fubini-Tonelli theorem: interchanging order of integration for measurable and L^1 functions on sigma-finite measure spaces
- Lebesgue's differentiation theorem: introduction and motivation
- Lebesgue's differentiation theorem: statement and proof - Part 1
- Lebesgue's differentiation theorem: statement and proof - Part 2
- DIfferentiation theorems: Almost everywhere differentiability for Monotone and Bounded Variation functions - Part 1
- DIfferentiation theorems: Almost everywhere differentiability for Monotone and Bounded Variation functions - Part 2
- Riesz's Rising Sun Lemma
- Differentiation theorem for monone continuous functions
- Differentation theorem for general monotone functions and Second fundamental theorem of calculus for absolutely continuous functions
- Live Session 20-11-2020